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A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n. 4
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 25, 49, 73, 121, 145, 217, 265, 361, 433, 553, 649, 817, 937, 1129, 1297, 1537, 1729, 2017, 2257, 2593, 2881, 3265, 3601, 4057, 4441, 4945, 5401, 5977, 6481, 7129, 7705, 8425, 9073, 9865, 373465, 374353, 738025, 1101865, 1828513 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..7000

FORMULA

a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289.

EXAMPLE

The a(10) = 25 matrices:

  [10]

.

  [4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]

  [2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2]

.

  [3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]

  [4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4]

.

  [2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]

  [4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4]

MAPLE

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)

      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))

    end:

a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n$2)):

seq(a(n), n=0..50);  # Alois P. Heinz, Jan 17 2019

MATHEMATICA

Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n, {k^2}], UnsameQ@@#&]], {k, n}], {n, 20}]

(* Second program: *)

q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,

     If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]];

a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]];

a /@ Range[0, 50] (* Jean-Fran├žois Alcover, May 20 2021 *)

CROSSREFS

Cf. A000009, A089299, A103198 (non-strict case), A120732, A323431, A323434, A323519, A323523, A323525, A323529.

Sequence in context: A283710 A247650 A090092 * A040601 A022359 A147491

Adjacent sequences:  A323519 A323520 A323521 * A323523 A323524 A323525

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 17 2019

STATUS

approved

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Last modified September 20 05:37 EDT 2021. Contains 347577 sequences. (Running on oeis4.)